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If the sum of the squares of the roots of the equation x2 - 14x + k = 0 is 100, then what is the value of k ?
- a)42
- b)48
- c)52
- d)56
Correct answer is option 'B'. Can you explain this answer?
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If the sum of the squares of the roots of the equation x2 - 14x + k = ...
Solution:
Given equation:
x^2 - 14x + k = 0
Sum of the squares of the roots:
Let the roots be α and β.
According to Vieta's formulas:
Sum of the roots, α + β = -(-14) = 14
Sum of the squares of the roots, α^2 + β^2 = (α + β)^2 - 2αβ
Given that α^2 + β^2 = 100, we have:
100 = 14^2 - 2αβ
100 = 196 - 2αβ
2αβ = 196 - 100
2αβ = 96
αβ = 48
Therefore, the product of the roots, αβ = 48.
Value of k:
In the given equation x^2 - 14x + k = 0, the product of the roots is equal to k.
So, k = αβ = 48
Therefore, the value of k is 48.
Hence, the correct answer is option (B) 48.
Given equation:
x^2 - 14x + k = 0
Sum of the squares of the roots:
Let the roots be α and β.
According to Vieta's formulas:
Sum of the roots, α + β = -(-14) = 14
Sum of the squares of the roots, α^2 + β^2 = (α + β)^2 - 2αβ
Given that α^2 + β^2 = 100, we have:
100 = 14^2 - 2αβ
100 = 196 - 2αβ
2αβ = 196 - 100
2αβ = 96
αβ = 48
Therefore, the product of the roots, αβ = 48.
Value of k:
In the given equation x^2 - 14x + k = 0, the product of the roots is equal to k.
So, k = αβ = 48
Therefore, the value of k is 48.
Hence, the correct answer is option (B) 48.
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Community Answer
If the sum of the squares of the roots of the equation x2 - 14x + k = ...
Given:
The sum of the squares of the roots of the equation x
2
- 14x + k = 0 = 100
Formula used:
For a quadratic equation ax2 + bx + c = 0
Sum of roots =
−
b
a
Product of roots =
c
a
Calculation:
x
2
- 14x + k = 0
α + β =
−
b
a
αβ =
c
a
Now, t
he sum of the squares of the roots
⇒ α2 + β2 = (α + β)2 - 2αβ
⇒ α
2
+ β
2
= (14)2 - 2k = 100 (Given)
⇒ 196 - 2k = 100
⇒ 2k = 96
⇒ k = 48
∴ The required value of k is 48.
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